For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts trivially), this is analytically the same as the space of one-dimensional representations of the fundamental group, i.e. (non-canonically) $LS(E)\cong \mathbb{C}^\times \times \mathbb{C}^\times$. This isomorphism is not algebraic, and I'm curious whether there is a nice description of $LS(E)$ rigid-analytically, when $E$ is an elliptic curve over $\mathbb{Q}_p$. (If convenient we can assume $E$ has a smooth model and has CM lifting the Frobenius -- though this shouldn't be necessary).

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universal vector extensionof $E$, and there is an old LNM by Mazur and (maybe?) Messing about it. It is a quasiprojective variety (which in particular has an underlying rigid analytic space). Regards, Matthew – Emerton Apr 29 '13 at 10:00